One CD player is said to have a signal-to-noise ratio of 82 dB, whereas for a second CD player it is 98 dB. What is the ratio of intensities of the signal and the background noise for each device?

The logarithmic ratio of intensities for both signal and noise for CD players gives the sound level of a CD player.

The signal to noise ratio for one CD player is \({\beta _1} = 82\;{\rm{dB}}\).

The signal to noise ratio for second CD player is \({\beta _2} = 98\;{\rm{dB}}\).

The ratio of intensities for both signal and noise for one CD player is calculated below:

\(\begin{aligned}{c}{\beta _1} = 10\log {\left( {\frac{{{I_{{\rm{signal}}}}}}{{{I_{{\rm{noise}}}}}}} \right)_1}\\{\left( {\frac{{{I_{{\rm{signal}}}}}}{{{I_{{\rm{noise}}}}}}} \right)_1} = {10^{\left( {\frac{{{\beta _1}}}{{10}}} \right)}}\end{aligned}\)

Here, \({I_{{\rm{signal}}}}\) and \({I_{{\rm{noise}}}}\) are the intensity of signal and noise.

Substitute the values in the above equation.

\(\begin{aligned}{c}{\left( {\frac{{{I_{{\rm{signal}}}}}}{{{I_{{\rm{noise}}}}}}} \right)_1} = {10^{\left( {\frac{{82\;{\rm{dB}}}}{{10}}} \right)}}\\{\left( {\frac{{{I_{{\rm{signal}}}}}}{{{I_{{\rm{noise}}}}}}} \right)_1} = 1.58 \times {10^8}\end{aligned}\)

The ratio of intensities for both signal and noise for second CD player is calculated below:

\(\begin{aligned}{c}{\beta _2} = 10\log {\left( {\frac{{{I_{{\rm{signal}}}}}}{{{I_{{\rm{noise}}}}}}} \right)_2}\\{\left( {\frac{{{I_{{\rm{signal}}}}}}{{{I_{{\rm{noise}}}}}}} \right)_2} = {10^{\left( {\frac{{{\beta _2}}}{{10}}} \right)}}\end{aligned}\)

Substitute the values in the above equation.

\(\begin{aligned}{c}{\left( {\frac{{{I_{{\rm{signal}}}}}}{{{I_{{\rm{noise}}}}}}} \right)_2} = {10^{\left( {\frac{{98\;{\rm{dB}}}}{{10}}} \right)}}\\{\left( {\frac{{{I_{{\rm{signal}}}}}}{{{I_{{\rm{noise}}}}}}} \right)_2} = 6.31 \times {10^9}\end{aligned}\)

Hence, the ratio of intensities for both signal and noise for first and second CD player are \(1.58 \times {10^8}\), and \(6.31 \times {10^9}\) respectively.